Is there way to move one vertex and make another move with it proportionally?
As pointed out by @MrZak you can use Shear (Ctrl+Alt+Shift+S)



Form the Blender Manual:


  Shearing is a form of movement where parallel surfaces move past one another. During this transform, movement of the selected elements will occur along the horizontal axis of the current view. The axis location will be defined by the Pivot Point. Everything that is “above” this axis will move (Shear) in the same direction as your mouse pointer (but always parallel to the horizontal axis). Everything that is “below” the horizontal axis will move in the opposite direction.